Modular $A_4$ Symmetry With Three-Moduli and Flavor Problem

TL;DR

This paper explores a modular $A_4$ symmetry framework with three moduli to address flavor problems, analyzing fixed points and residual symmetries for leptons and quarks, and demonstrating compatibility with experimental data.

Contribution

It introduces a three-moduli modular $A_4$ symmetry model with different moduli assignments for fermions, analyzing fixed points and residual symmetries to explain flavor structures.

Findings

Lepton masses and mixings match experimental data under certain conditions.

Quark masses and mixings are correctly generated near fixed points.

A small deviation from fixed points is necessary for non-trivial right-handed neutrino assignments.

Abstract

The modular $A_4$ symmetry with three moduli is investigated. We assign different moduli to charged leptons, neutrinos, and quarks. We analyze these moduli at their fixed points where a residual symmetry exists. We consider two possibilities for right-handed neutrinos. First, they are assumed to be singlets under modular symmetry. In this case, we show that the lepton masses and mixing can be obtained consistently with experimental observations. Second, they are assigned non-trivially under modular symmetry. We emphasize that a small deviation from their fixed point is required in this case. Finally, the quark masses and mixing are generated correctly around the fixed point of their modulus. In our analysis, we only consider the simple case of weight 2.